The invention is based on a fuel injection system of a general type described hereinafter. A fuel injection system having an electromagnetically actuable valve is already known, the valve having a feed line to a supply line having fuel flowing through it, with a return flow line being disposed in the feed line by way of which excess fuel can flow back to the supply line. However, this apparatus has the disadvantage that the fuel which has been warmed in the injection valve and may now contain vapor bubbles is delivered to the next subsequent injection valve and there causes unsatisfactory fuel injection, which may be expensive when the engine comes to a stop.